Let $\Lambda \subset {ℝ}^n\times {ℝ}^m$ and $k$ be a positive integer. Let $f:{ℝ}^n\to {ℝ}^m$ be a locally bounded map such that for each $(\xi ,\eta )\in \Lambda$, the derivatives $D_{\xi }^{j}f\left(x\right):=\frac{d^j}{d{t}^j}f(x+t\xi ){|}_{t=0}$, $j=1,2,\cdots k$, exist and are continuous. In order to conclude that any such map $f$ is necessarily of class $C^k$ it is necessary and sufficient that $\Lambda$ be not contained in the zero-set of a nonzero homogenous polynomial $\Phi (\xi ,\eta )$ which is linear in $\eta =({\eta }_1,{\eta }_2,\cdots ,{\eta }_m)$ and homogeneous of degree $k$ in $\xi =({\xi }_1,{\xi }_2,\cdots ,{\xi }_n)$. This generalizes a result of J. Boman for the case $k=1$. The statement and the proof of a theorem of Boman for the case $k=\infty$ is...

Let $A_1,\cdots ,A_m$ be $n\times n$ real matrices such that for each $1\le i\le m,$ $A_i$ is invertible and $A_i-A_j$ is invertible for $i\ne j$. In this paper we study integral operators of the form $$Tf\left(x\right)=\int k_1(x-A_{1}y)k_2(x-A_{2}y)\cdots k_m(x-A_{m}y)f\left(y\right)\mathrm{d}y,$$ $k_i\left(y\right)={\sum }_{j\in \mathbb{Z}}{2}^{jn/q_i}{\varphi }_{i,j}\left(2^{j}y\right)$, $1\le q_i<\infty ,$ $1/q_1+1/q_2+\cdots +1/q_m=1-r,$ $0\le r<1,$ and ${\varphi }_{i,j}$ satisfying suitable regularity conditions. We obtain the boundedness of $T:H^p\left({ℝ}^n\right)\to L^q\left({ℝ}^n\right)$ for $0<p<1/r$ and $1/q=1/p-r.$ We also show that we can not expect the $H^p$-$H^q$ boundedness of this kind of operators.

We deal with the integral equation $u\left(t\right)=f\left({\int }_{I}g(t,z)u\left(z\right)dz\right)$, with $t\in I=[0,1]$, $f:{𝐑}^n\to {𝐑}^n$ and $g:I\times I\to [0,+\infty [$. We prove an existence theorem for solutions $u\in L^{\infty }(I,{𝐑}^n)$ where the function $f$ is not assumed to be continuous, extending a result previously obtained for the case $n=1$.

We study the realization $A$ of the operator $\mathcal{A}=\frac{1}{2}\mathrm{△}-(DU,D\cdot )$ in $L^2\left(\mathrm{\Omega },\mu \right)$, where $\mathrm{\Omega }$ is a possibly unbounded convex open set in ${\mathbb{R}}^N$, $U$ is a convex unbounded function such that ${lim}_{x\to \partial \mathrm{\Omega },x\in \mathrm{\Omega }}U\left(x\right)=+\mathrm{\infty }$ and ${lim}_{\left|x\right|\to +\mathrm{\infty },x\in \mathrm{\Omega }}U\left(x\right)=+\mathrm{\infty }$, $DU\left(x\right)$ is the element with minimal norm in the subdifferential of $U$ at $x$, and $\mu \left(dx\right)=c\exp \left(-2U\left(x\right)\right)dx$ is a probability measure, infinitesimally invariant for $\mathcal{A}$. We show that $A$, with domain $D\left(A\right)=\left\{u\in H^2\left(\mathrm{\Omega },\mu \right):\left(DU,Du\right)\in L^2\left(\mathrm{\Omega },\mu \right)\right\}$ is a dissipative self-adjoint operator in $L^2\left(\mathrm{\Omega },\mu \right)$. Note that the functions in the domain of $A$ do not satisfy any particular boundary condition. Log-Sobolev and Poincaré inequalities...